Introduction
The objective of this work is to identify the parameters of a hyperelastic constitutive law of an artery sample that is tested for inflation as in the experiment illustrated in the figure below
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Data of the problem
The displacement field is measured with stereo image correlation during the inflation test.
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Data of the problem
• A cloud of 644 points is defined on the zone of the field of view in initial position which is defined as the reference configuration. The
coordinates in the reference configuration are (x_0, y_0, z_0). For each point, the image stereo-correlation method can reconstruct the
coordinates (x, y z) for 43 loading steps. Each loading step corresponds to a pressure P applied in the system.
• The problem data is available in Matlab format in the data_TP.mat file.
You will find there (x_0, y_0, z_0), (x, y z) and P.
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Mechanical response
• Plot the curve of variation of P as a function of the maximum value of z.
• The opposite problem consists in finding a constitutive law for the membrane which would reproduce the
same behavior curve. One solution would be to
develop a finite element model of the swelling test and then readjust the model parameters so as to minimize the difference between the displacements measured and the displacements predicted by the model.
• We will develop alternative approaches based on the VFM.
Derivation of local basis
• We define a triangular mesh from the point cloud (x_0, y_0, z_0), each point being a node. The
connectivity matrix of this mesh is given in the form of a Matlab matrix called TRI.
• We will reconstruct at the center of gravity of each triangle the gradient of the transformation and the Green Lagrange deformations from the available data. They are provided in DEF_GRADIENT.mat and EPSILON.mat if necessary.
• The code is provided (calcul_F.m) as well as the results if necessary in Matlab matrices. Execute calcul_F.m.
Main variables generated by calcul_F.m
• xGdeb, yGdeb, zGdeb: coordinates of the centers of gravity of the triangles in the initial state
• xGfin, yGfin, zGfin: coordinates of the centers of gravity of the triangles in the initial state
• vect3: vector normal to the surface in the initial state
• vect3b: vector normal to the surface in the final state
• F_tensor: contains F11, F22, F12 and F22 (components of the gradient of the transformation) for each triangle and each pressure value
• E_tensor: contains E11, E22, E12 (components of the strains of Green Lagrange) for each triangle and each value of pressure
Normal vectors
• Why did you calculate the normal vectors at the surface?
• From the norms, one can define a local coordinate system of the membrane then
calculate the gradient of the transformation and the strains of Green Lagrange.
• How are normal vectors calculated?
• We compute the normal vectors at the surface at each node from the data. This is done by fitting the surface locally by a quadric. Matlab code is provided: initial_normals.m and final_normals.m
Normal vectors
Strain fields
• Graphically represent the fields obtained (E11, E22, E12,… at different pressure values) in color maps using the function provided (Myplot.m).
What do we see?
Stress fields
• Suppose that the behavior of the tissue has a NeoHookean hyperelastic behavior and that its hyperelastic constant is 1 MPa.
• Find the expression of the stress as a function of the strain for an incompressible NeoHookean material.
• Reconstruct the Cauchy stress field in the local coordinate system. To check whether the constraints obtained are at equilibrium, we will write the principle of virtual powers at each loading step.
I c F
. F C
2 10 t
Fields of virtual fields
• The virtual velocity field that is proposed is a swelling field.
• Each point in the tissue sample is displaced by virtually the same amount in the normal
direction.
• To use calcul_def_virt.m to find the field of virtual strain rates at each stage of loading. These virtual strain rates are reconstructed in the deformed
local coordinate system (Eulerian expression).
They are provided in EPSILON_VIRT.mat if necessary.
Principle of virtual power
• Reconstruct the interior virtual power vector over the entire measurement area.
• Reconstruct the vector of the power of the external forces by showing that it reduces to the virtual power of the forces of pressure.
• Trace the variations of these virtual powers (interior and exterior) according to the
different stages of loading.
Identification of a hyperelastic constant
• Set up a method of least squares to identify the hyperelastic constant of the NeoHookean model.
• This amounts to performing a regression on
the external virtual power curve by an internal virtual power curve depending on the
constant to be determined.
• After identifying the constant, calculate the
coefficient of determination R2.
Identification of a hyperelastic
constant
Identification of other hyperelastic models
• In order to improve R2, perform the
identification with other more suitable
hyperelastic behavior laws (for example, Yeoh, Ogden, Holzapfel). Set up the least squares
regression for one of these models.
Spatial variations of hyprelastic constants
• Show that the identification method just implemented can be applied to any area of the sample to reconstruct the spatial
distribution of hyperelastic constant.
• Under which assumptions is the developed method valid? Under what conditions is this assumption likely to be called into question?
What to do if this assumption is not valid.
To go further…
Davis et al. Biomechanics and Modeling in Mechanobiology - 2015.
